$[30$pt$]$ Consider the two$-$dimensional grid $)=(y$ representing steady$-$state conditions with a uniform heat generation at a volumetric rate $q^{}$ for a system with thermal conductivity $k.$ The width $($thickness$)$ of the system is $w=x($into the paper$).$

a$.[20$pt$]$ Using the energy balance method, derive the finite$-$difference equations for node $6,$ in terms of $T_2,T_5,T_9,T_7,$ and $q^{}.$

b$.[10$pt$]$ Using the energy balance method, derive the finite$-$difference equations for node $7,$ in terms of $T_3,T_6,q^{},h,$ and $T_{}.$